Sir, I am obliged to your correspondent "." for referring me to the work where Mr. Lucy's improvements on the steam-engine is fully described. It has no fly-wheel whatever, it appears, but in place of it there is a cogged-wheel made fast on the crank shaft, the diameter about twice the length of the throw of the crank; and this wheel takes a smaller wheel, made fast to the axle of a second crank, which latter is one half the dimensions of the former, so that two revolutions of the large wheel will make the small wheel give four. Farther, there is a connecting rod and beam to the second crank, and a cylinder and piston rod at the other extremity of the same, which cylinder has a bottom but no top, and is the pneumatic apparatus for equalizing the motions, for which purpose the arrangement is admirably contrived. A little consideration will show, that the pressure of the atmosphere acting on the piston of the second cylinder, will make the main crank pass the centre by a simpler arrangement than would be required with a double engine; and I am surprised that your correspondent should have expressed himself in the manner he has done, relative to this improvement, which appears to me to be one of greater consequence than any which has been made in the crank engine for many years. Your correspondent admits that Mr. Parkes's authority, as far as the stating of facts is concerned, is unquestionable; the same also with respect to Mr. Scott Russell. Now the latter gentleman admits that the increased work of each pair of stones was in the proportion of 56 to 52, and Mr. Parkes states that an additional pair had been put up in the mill. And these, be it observed, are not inferences but naked facts; and your correspondent himself admits that the friction with the new contrivance must be, as no doubt it is, considerably greater than the friction with the flywheel. Why then, let me ask, can he doubt the inference drawn by Mr. Parkes from these data, that the power of the engine is considerably increased, and fully to the extent of 11 per cent? But your correspondent contends, that if such be a fact it is incumbent on Mr. Parkes to prove (as the increased power, he says, could come from no other quarter), that there must be less friction in the new apparatus than there was with the old fly-wheel. I do not consider this a very logical conclusion, for it amounts to this, that unless he proves an impossibility, his facts must go for nothing. Mr. Parkes, according to his notion, must connect his facts with your correspondent's conclusions, while at the same time he will not connect his mathematical conclusions with the facts which he admits are undeniable. He expresses himself also in a strange way, with respect to the statement made by Mr. Scott Russell; as for instance, that he is surprised he should have stated such and such facts, doubts whether he knew the bearings of them, that it is to be regretted that he did so, &c. Why, did your correspondent expect that Mr. Russell would have acted so disingenuously and improperly, as to be deterred from stating facts even had they gone counter, as they have, in this instance, to his own published opinions? One example of the mode in which your correspondent draws erroneous conclusions from facts worth nothing is particularly deserving of notice. doubt a hundred apples purchased at the market will not, with any newly discovered basket deliver 111 apples at home; but he concludes from this that another hundred apples sent home in another description of basket, could not deliver less than the hundred. With holes in the basket, for instance, there might be several lost, and but 80 delivered. The analogy is anything but complete; mathematicians are not bound to connect mathematical facts with practical ones; but practical facts, according to his views, must be proved not to vary from mathematical ones; while at the same time it is admitted, that all the elements connected with the practical fact are not embraced in the other case. In conclusion, I should be glad to know your correspondent "M's." opinion on Mr. Lucy's improvements; I mean whether he would think there is any loss by such an arrangement of crank move ON THE CONSTRUCTION AND USE OF COMMUTATION TABLES, FOR CALCULATING THE VALUES OF BENEFITS DEPENDING ON LIFE CONTINGENCIES. [We find it will be more convenient, instead of the notation proposed in our first paper, (p. 428,) to represent the indications of the table of mortality, to make use of the following, which will therefore be adhered to in the remainder of these papers. We shall employ, with the age attached, not as a suffix, but enclosed in parentheses, to denote the number who attain each age. Thus, the numbers represented by the table to attain the ages 0, 1, 2, 3,...... x years, will be respectively denoted by 1(0), 4(1), 7(2), (3), ... b (x). And hence, the number who die in their 1st year will be denoted by 1(0) 7(1) Part II.-Description and We propose in the present paper to describe the Commutation Tables, explain their construction, and show the algebraical properties that belong to them in virtue of that construction. We refer for illustration to the Table contained in the two following pages. The Table consists, it will be seen, of two side columns containing the ages, and five inner columns, headed D, N, S, M, and R, respectively. There is another column headed C, which is essential to the theory of the tables, and the proper place for which, if inserted, would be between columns S and M. But as this column is not required in practice, it is never exhibited. The letters D, N, S, &c., are chosen quite arbitrarily, and have no reference to the signification of the columns at the head of which they are placed. The number in any column, opposite to any age, is denoted by the letter at the head of the column, with the age attached, enclosed in parentheses. Thus, the number corresponding to age 20 in column D, for example, is denoted by D (20), in column N, by N (20), and so of the other columns. If the age be x, N (20) = D (21) + D (22) 1(2) (x-1) 1(3) 1(x) (x) — 1(x+1) and so on.] Properties of the Tables. the corresponding numbers are denoted by D (x), N (x), &c. The columns are constructed as follows: Take any age, 20, for example. The number in column D, opposite age 20, is equal to the product of the number represented by the mortality table to attain the age 20, into the present value of one pound due at the end of 20 years. So that the algebraical expression will be, D (20) = (20) v2o. And generally, the number corresponding to any age in column D, is equal to the number who complete that year of their age, multiplied by the present value of one pound due at the end of as many years as are equal to the age. Hence, the general expression for the value of the numbers in column D will be, D (x) = l (x) v*,* r denoting any age. Hence also, if the age be 0, we have, D (0) = 1 (0) vo = 7 (0), since vo = 1. That is, the first number in column D is equal to the radix of the Mortality Table. Column N is formed from column D, by inserting opposite each age in N the sum of the numbers opposite all the higher ages in D. For example, + D (23) + &c. to the end. N (21) = D (22) + D (23) + D (24) + &c. So that the expression for the general term of this column is N (x) = D(x + 1) + D (x + 2) + D (x+3)+ &c. x, as before, denoting any age. It follows from this that the last N in the Table is 0.1 For, since N (x) = D (x + 1) + . . . ., if x be the highest age, D(+1) and all the following terms vanish, since there are no survivors at the ages denoted by x+1, &c. Also, the first N in the Table is equal to the sum of all the D's except the first; or N (0) = D (1) + D (2) + D (3) + Column S is formed by inserting in it, opposite each age, the sum of the numbers in N, opposite that age and all the higher ages. The expression for the general term is, therefore, S(x)=N (x) + N (x + 1) + N(x + 2) + &c. By an expression for the general term of a series, is meant, an expression in which a variable quantity is introduced, and which, by giving any particular value to the variable, gives the term of the series corresponding to that value. Thus, in the above general expression, D(r) = 1(x)vx, x, which denotes the age, is the variable; and if we give to it a particular value, we have immediately the term of the series corresponding to that value. For instance, if x 20, the expression becomes, D (20) 7 (20) x20; and this is the value in column D corresponding to age 20. And it differs from D in including, at each age, the number corresponding to that age in the preceding column. It follows from this, that the last S is 0, and that the first S is equal to the sum of all the numbers in column N. The three columns just described are called the annuity columns. The remaining three are called the assurance columns. The last named we now proIceed to describe. Column C is formed by inserting opposite each age, the product of the number who die in the following year of M (x) = C (x) + C (x Column R, also, is formed from M, in exactly the same manner as M was R (x) = M (x) + M (x + We proceed now to demonstrate a few of the algebraical properties, and relations amongst each other, that belong to the numbers in the different columns. These will be useful for after reference, with a view to which we shall number them as we go along. By algebraical properties and relations we mean those properties and relations D (x) = (x) va C (x) = [l(x) − 1 (x + C (1) = [7 (1) — 7 (2)] v2, and so on. And the expression for the general term will be C (a) = [ l (x) − 1 ( x + 1 ) ] v*+1 Column C, as already remarked, is formed from C; and hence the expres- that subsist in virtue of the mode in which Among the algebraical relations must be classed those which we have just seen to subsist; which, therefore, we here repeat, for the convenience of reference. N (x) = D (x + 1) + D (≈ + 2) + D (x + 3) +, &c. = &c. &c. } (1.) (2.) that it shall not exceed the difference between a and the oldest age in the Table, so that x + n may not exceed that age. Making this substitution, then, the above equations will still hold, and we shall have In what follows, we call x the present age. And since, in the above expressions, a denotes any age, we may substitute for it + n, the only limitation† that we make with regard to n being, D(x + n) = l (x + n) v * + n C (x + n) = [1 (x + n) − 1 (x + n + 1] v2 + n + 1 N (x + n) · D (x + n + 1) + D (x + n + 2) +, S(x + n) = N (x + n) + N (x + n + 1) +, &c. . M (x + n) C (x + n) + C (x + n + 1) +, &c. R (x + n) = M (x + n) + M (x + n + 1) +, &c. It thus appears, that the nth term from the present age, in columns S, M, and R, expresses the sum of the nth and following terms from the same age, in columns N, C, and M, respectively; and that the nth term from the present age, = N (xn−1) = D (x + n) + * It will be afterwards seen, that there is no necessity for employing those particular powers of v, which have been made use of in the construction of the table. The only condition as to these powers, which is indispensable to the possession by the table of the required properties, is, that their indices shall form an increasing arithmetical series, &c. in column N, expresses the sum of all the terms after the nth, from that age in column D. If we wish the sum of the nth and following terms in D, we have only to write a 1 for a in the first of the expressions (4), which then becomes, D (x + n + 1) +, &c. (5.) of which the common difference is unity. The powers that have been chosen possess the advantage of imparting to the expressions for the values of the numbers in columns D and C, a symmetry that would not otherwise have belonged to them. +Algebraically considered, it is not necessary to make this limitation. |